Is this statement true?

Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.

If $\varphi$ is polynomially growing and of infinite order, then the semidirect product $\mathbb{F}\rtimes_\Phi\mathbb{Z}$ is acylindrically hyperbolic but not relatively hyperbolic.

If there is anything unclear about notation, please tell me. This statement is closely related to Problem 8.2 in *A. Minasyan and D. Osin. Acylindrical hyperbolicity of groups acting on trees.
math. Annalen, 362:1055–1105, 2015* (arXiv link). This problem asks which mapping tori of injective endomorphisms of finitely generated free
groups are acylindrically hyperbolic.